Average Error: 34.7 → 10.1
Time: 6.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r61394 = b;
        double r61395 = -r61394;
        double r61396 = r61394 * r61394;
        double r61397 = 4.0;
        double r61398 = a;
        double r61399 = r61397 * r61398;
        double r61400 = c;
        double r61401 = r61399 * r61400;
        double r61402 = r61396 - r61401;
        double r61403 = sqrt(r61402);
        double r61404 = r61395 + r61403;
        double r61405 = 2.0;
        double r61406 = r61405 * r61398;
        double r61407 = r61404 / r61406;
        return r61407;
}
↓
double f(double a, double b, double c) {
        double r61408 = b;
        double r61409 = -1.2447742914077108e+109;
        bool r61410 = r61408 <= r61409;
        double r61411 = 1.0;
        double r61412 = c;
        double r61413 = r61412 / r61408;
        double r61414 = a;
        double r61415 = r61408 / r61414;
        double r61416 = r61413 - r61415;
        double r61417 = r61411 * r61416;
        double r61418 = 6.485606601696406e-71;
        bool r61419 = r61408 <= r61418;
        double r61420 = -r61408;
        double r61421 = r61408 * r61408;
        double r61422 = 4.0;
        double r61423 = r61422 * r61414;
        double r61424 = r61423 * r61412;
        double r61425 = r61421 - r61424;
        double r61426 = sqrt(r61425);
        double r61427 = r61420 + r61426;
        double r61428 = 1.0;
        double r61429 = 2.0;
        double r61430 = r61429 * r61414;
        double r61431 = r61428 / r61430;
        double r61432 = r61427 * r61431;
        double r61433 = -1.0;
        double r61434 = r61433 * r61413;
        double r61435 = r61419 ? r61432 : r61434;
        double r61436 = r61410 ? r61417 : r61435;
        return r61436;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2447742914077108e+109

    1. Initial program 49.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 4.0

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified4.0

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

    if -1.2447742914077108e+109 < b < 6.485606601696406e-71

    1. Initial program 13.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv13.6

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 6.485606601696406e-71 < b

    1. Initial program 53.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 8.4

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))